一个质量为 $m$ 的质点 $P$ 连接在一根自然长度为 $a$、弹性模量为 $\frac43mg$ 的轻质弹性细绳的一端。细绳的另一端连接在粗糙水平面上的固定点 $O$。质点静止在该平面上，细绳处于自然长度。$P$ 与该平面之间的摩擦系数为 $\frac13$。质点以速率 $\frac12\sqrt{ga}$ 沿该平面向 $OP$ 方向射出。

求随后运动过程中细绳的最大伸长量。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

Where a numerical value for the acceleration due to gravity ($g$) is needed, use $10\,\mathrm{m\,s^{-2}}$.

需要使用重力加速度 $g$ 的数值时，取 $10\,\mathrm{m\,s^{-2}}$。

伸长达到最大时，质点第一次瞬时静止。初动能转为弹性势能与克服摩擦力所做的功。

Let $x$ be the maximum extension. The friction has magnitude $mg/3$ while the particle moves away from $O$. Using work and energy,

Cancelling $mg$ gives

Multiplying by $24a$ and rearranging gives

Since the extension is positive, its maximum is $a/4$. During this outward interval both tension and friction oppose the motion, so this is the first stop. Here the tension is $(4mg/3a)(a/4)=mg/3$, which static friction can balance. The particle can therefore remain at rest, and this is its greatest extension.